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An Abstract Approach to Polychromatic Coloring: Shallow Hitting Sets in ABA-free Hypergraphs and Pseudohalfplanes

  • Balázs Keszegh
  • Dömötör  PálvölgyiEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9224)

Abstract

The goal of this paper is to give a new, abstract approach to cover-decomposition and polychromatic colorings using hypergraphs on ordered vertex sets. We introduce an abstract version of a framework by Smorodinsky and Yuditsky, used for polychromatic coloring halfplanes, and apply it to so-called ABA-free hypergraphs, which are a generalization of interval graphs. Using our methods, we prove that \((2k-1)\)-uniform ABA-free hypergraphs have a polychromatic k-coloring, a problem posed by the second author. We also prove the same for hypergraphs defined on a point set by pseudohalfplanes. These results are best possible.

We also introduce several new notions that seem to be important for investigating polychromatic colorings and \(\epsilon \)-nets, such as shallow hitting sets. We pose several open problems related to them. For example, is it true that given a finite point set S on a sphere and a set of halfspheres \(\mathcal {F}\), such that \(\{S\cap F\mid F\in \mathcal {F}\}\) is a Sperner family, we can select an \(R\subset S\) such that \(1\le |F\cap R|\le 2\) holds for every \(F\in \mathcal {F}\)?

Keywords

Interval Graph Color Classis Dual Version Pseudoline Arrangement Unbounded Convex 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgement

We would like to thank the anonymous referees for their several useful suggestions and comments.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Alfréd Rényi Institute of MathematicsBudapestHungary
  2. 2.Institute of MathematicsEötvös UniversityBudapestHungary

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